Pregroups and Bass-Serre Theory

86 Pages·1987·7.105 MB·English

by  Frank Rimlinger

Memoirs of the American Mathematical Society
Number 361
Frank Rimlinger
Pregroups and
Bass-Serre theory
Published by the
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island, USA
January 1987 • Volume 65 • Number 361 (fourth of 5 numbers)
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Table of Contents
Introduction v
Part I: Stallings's theorem about pregroups;
introduction to pregroups of finite height
1. The definition of a pregroup 1
2. The pregroups of finite height 5
3. The subpregroup of units 13
Part II: A presentation for the universal group
of a pregroup of finite height.
4. Pregroup actions and generating sets 20
5. A presentation for the universal group of a
pregroup of finite height 29
Part III: The relationship between pregroups
and graphs of groups
6. A graph of groups structure for pregroups
of finite height 39
7. A pregroup structure for graphs
of groups of finite diameter 55
References 73
in
Abstract
This paper investigates the structure of pregroups, Stallings generalization of free
products with amalgamation. Stallings discovered an order relation on pregroups
which gives rise to the concept of finite height. We show that the universal group of
a pregroup P of finite height may be realized as the fundamental group of a graph of
groups. The vertex groups of this graph of groups correspond to the P-conjugacy
classes of the maximal subgroups of P. Conversely, we construct a pregroup struc
ture for the fundamental group of any graph of groups whose geodesies are of finite
bounded length. Via a theorem of Karrass, Pietrowski, and Solitar, we deduce that a
group is a finite extension of a finitely generated free group if and only if it is the
universal group of a finite pregroup.
1980 Mathematics Subject Classification. Primary 20E07, 20E34, 20F10; Secondary 20E06.
Keywords and phrases. Pregroup, word problem, reduced word, graph of groups, HNN extension,
free by finite.
Library of Congress Cataloging-in-Publication Data
Rimlinger, Frank, 1957-
Pregroups and Biss-Serre theory.
(Memoirs of the American Mathematical Society, 0065-9266;
no. 361 (Jan. 19S7»
"January 1987, volume 65, number 361 (fourth of 5 numbers).'
Bibliography: p.
1. Pregroups. I. Title. II. Title: Bass-Serre theory. III. Series.
QA3.A57 no. 361 [QA171] 510s [512'.22] 86-32112
ISBN 0-8218-2421-X
IV
Introduction
Free groups, free products with amalgamations, and HNN extensions may be
thought of as "groups defined by reduced words." We make this concept precise as
follows: Let G be a group, and let P c G be a subset of G such that P generates G
and P=P~l. Let D CPxP be a subset of pairs (x,y) of elements of P such that for
all (x,y)eD, xy eP. A word (x , ... ,x )ePn is said to be reduced if for each adja
L n
cent pair (x ,x ) it is not the case that {x x )eD, (for otherwise we reduce the
z / + 1 h i + [
word in the obvious way). We sa^ that (P,D) is a reduced word structure for G if for
all g EG all reduced words representing g are of the same length.
In Rimlinger [to appear] we proved that if (P,D) is a reduced word structure for
G, then (P,D) is a pregroup and G is isomorphic to U(P), the universal group of P.
But what is a pregroup? These objects were created by John Stallings [1971], who
traces his idea back to van der Waerden [1948] and Baer [1950]. In Stallings treat
ment, a pregroup is a set P, together with a subset D CPxP, and a special element
1 eP, called the identity element. In addition, pregroups are endowed with a partial
multiplication rn:D-+P and an involution i:P^>P. Moreover, certain axioms con
cerning the above sets and maps must also hold. These axioms may be paraphrased
by saying that 1 is the identity element, m:D->P is "as associative a multiplication
as possible," and i:P-+P takes an element to its inverse. Additionally, if three pairs
(w,x), (x,y), and (y,z) are in Z>, then either (wx,y)eD or (xy,z)eD.
Stallings defined the universal group U(P) of a pregroup (P,D) to be the free
group on the set P modulo the relations { xy=z \ x,y,zeP and z=rn(x,y) }. If
G is a group isomorphic to U(P), we say, somewhat loosely, that G has a pregroup
structure P. Stallings original theorem about pregroups, interpreted in the language
developed above, is that (P,D) is a reduced word structure for U(P). Thus pre
groups exactly capture the notion of "groups defined by reduced words," in that for
any group G, (P,D) is a reduced word structure for G if and only if (i) (P,D) is a
pregroup, (ii) G and U(P) are isomorphic, and hence (iii) P is a pregroup structure
for (7.
For example, if P= { x,x_1,l }
5
D={ (x,x-l)Xx-\xUUx)XxM(hx-l)Xx-\\WA) }
and rn.D-^P and i:P-+P are defined in the obvious way, then (P,D) is in fact a
pregroup, and U(P) is a free group of rank 1. The reduced words with respect to D
representing elements of U(P) are obtained from arbitrary words in the alphabet P
via free reduction.
v
VI RIMLINGER
Stallings showed that free products with amalgamation and HNN extensions
also have pregroup structures. It is a fact that if (P,D) is a pregroup, QCP,
QxQcD, and (x,y)eQxQ =^>xyeQ, then Q inherits a group structure from P.
In this event we say that Q is a subgroup of P. For example, in Stallings's original
constructions of pregroup structures for A * B and v_") , the maximal subgroups
C r
of the pregroup structures are A and B in the case of the free product with amalga
mation and A in the HNN case.
Two natural questions arise at this point. Given a pregroup P, does U(P) act
on a tree? Conversely, given a group which acts on a tree, does this group have a
pregroup structure which reflects the structure of the action. We answer both these
questions in the affirmative, provided (i) the tree in question is an ordinary simpli-
cial tree, (ii) the quotient graph of the tree in question has geodesies of finite
bounded length, and (iii) the pregroup in question satisfies pregroup theoretic cri
teria analogous to (i) and (ii). The exact results are stated here. The summary
below refers the pregroup theoretic notions to the main body of the paper.
Theorem A: Let P be a pregroup of finite height . Let G be the union of the fun
damental group systems for each Ul{P\ i=0, . . . ,d. Then P is a pregroup struc
ture for the fundamental group of a graph of groups ^(F^F). The base groups of Y
are in 1-1 correspondence with the elements of G, and the corresponding groups are
isomorphic. The graph Y is such that the oriented edges of the complement of a
maximal tree in Y are in 1-1 correspondence with the union of the spanning sets of
the fundamental groups of G.
Theorem B: Let (H,7) be a graph of groups with bases V, edges E, basepoint v ,
0
and maximal tree T. Suppose the graph Y is of finite diameter and (H,7) is proper.
Let XcE be the edges of Y not in T, and let X+ be an orientation of X. Then
TTI(H,Y,VQ) has a pregroup structure Q satisfying the following conditions:
(i) for some d > 0, Q has depth d, and a good Q-sequence
((G ,0), . . . , (G 0) (G , X)) satisfying (ii), (iii), and (iv):
d ]? 5 o
(ii) for i = \, . . . ,d, the groups of G, are in 1-1 correspondence with the maxi
mal bases of depth i and corresponding fundamental groups and base
9
groups are isomorphic,
(iii) X + is in 1-1 correspondence with X+,
(iv) G is in 1-1 correspondence with
0
{veB | depth(v)=0 }(J B, where ^ C { veB \ depth(v)>0}.
If G eG corresponds to G for some v eB of depth 0, then G is isomorphic
0 y
to H eH. If G GG corresponds to some veB, then G is isomorphic to a
v 0
subgroup of H. •
v
PREGROUPS vn
From theorems A and B and result of Karrass, Pietrowski, and Solitar [1973]
generalizing Stagings' work on the ends of a group [1968], we deduce the following
corollary.
Corollary: A group G is free by finite if and only if G is the universal group of a
finite pregroup.
The paper is organized as follows:
Part I: We review Stallings' work on pregroups and the connection between pre-
groups and groups defined by reduced words. We make an initial investigation of
the subgroups of a pregroup. We provide a geometric interpretation of the pregroup
axioms, which serves as a convenient computational tool for part II.
We exploit a tree ordering, discovered by Stallings, on the elements of a pre
group in order to define the subcategory of pregroups of finite height. We define the
full subpregroup of units, U(P), of a pregroup P, and show how this leads to a des
cending sequence PDU(P)DU\P) • * • D Ud(P) of subpregroups of given pregroup
P of finite depth d.
Part II: We define the notion of a pregroup action on a set. We show that the
subpregroup of units U(P) acts on the maximal vertices of P. We exploit this action
to define a generating set for P, consisting of a fundamental group system and a
spanning set for P. We prove that in a certain sense U(P)\J{ fundamental group
system } U{ spanning sets } is a subset of P which minimally generates U(P),
(theorem 4.24). This result pares the way for theorem 5.3, in which we give a
presentation of P in terms of a generating set of P. We give an example which illus
trates this presentation and shows how to get from a pregroup to a graph of groups.
As an easy application of this presentation, we prove the the universal group of a
finite pregroup with no nontrivial subgroups is free.
Part III: We give an example which indicates the major technical problem of the
proof of theorem A. This example motivates an inductive argument based on the
presentation of part II. To prove theorem B, we first review chapter one of Serre
[1980] from a pregroup theoretic point of view. We construct a pregroup structure,
due to Stallings, for F{H,Y). This group is defined in Serre [1980] and is a large
group containing ^^11,7,v ), the fundamental group of a graph of groups. We
0
define, for each aeF(H,Y\ a sequence of paths 7(a) in 7 reflecting the reduced
word structure for a, which in turn is derived from the pregroup structure for
F(H,7). Using this idea we define a subset Q of F(H,Y) and prove that Q is in
fact a subpregroup of F(H,7). Finally, by reference to the proof of theorem A, we
show that U(Q) is isomorphic to 7r{H,Y,V ), establishing theorem B. We end the
{ 0
paper with examples showing how to go from a free product with amalgamation or
an HNN extension to a pregroup. These examples resemble some of the original
RIMLINGER
Vlll
examples of Stallings [1971].
I would like to thank John Stallings for helping me get started in the pregoup
business, and for many valuable suggestions. In particular, the suggestion that pre-
groups were somehow related to Bass-Serre theory set me going in the right direc
tion.
Parti
Stallings's theorem about pregroups;
introduction to the pregroups of finite height
1. The definition of a pregroup
In this section we review some basic facts about pregroups. We define reduced
word structures and pregroup structures. By theorem 3, theorem 4 and corollary 7
below, these concepts are equivalent. References are given for proofs of theorems 3
and 4, but otherwise the exposition is self contained. Anticipating results to come,
we end this section with a discussion of subgroups of a pregroup.
We start by considering prees, or sets with partial multiplication table. Let P
be a set, let D cPxP, and let m\D-+P be a set map. Typically we denote m(x,y)
by xy or x*y. The composite concept (P,D,m) is called a pree. The universal
group V{P) of a pree P is determined by the presentation
(P;{ m(x,y)y~lx~l | (x,y)eD }). The prees form a category C. Given prees
(P,D) and (Q,E), we define [P,Q] to be the set maps fcP^Q such that
C
(x,y)eD =^>((j){x),(t)(y))eE and ^(xy)=(j>(x)())(y). Clearly there is a forgetful functor
F:G->C from groups to prees, and if G is a group, then the diagram
U(P)
I
13!
I
G
shows that [U(P),(J]G and [P,F(G)]C are in 1-1 natural correspondence. The map
L:P-+U(P) is the restriction to P of the natural projection F(P)-+U(P) from the free
group on the set P onto U(P). Let (P,D) be a pree. Here is some useful terminol
ogy.
P-word X of length n: An element X = {x . . . ,x )ePn
h n
X is P-reduced:\his means V 1 < *" < fl -1, (x x i)£D.
h i +
X represents x: This means X=L(XI)L(X) * * • t(x„)eU(P).
2
Recieved by the editor July 14, 1985.
*
The author was a Sloan Foundation Doctoral Dissertation Fellow at UC Berkeley during the period
this paper was written.
1